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December, 2019 Enumerations of Permutations by Circular Descent Sets
Hungyung Chang, Jun Ma, Jean Yeh
Taiwanese J. Math. 23(6): 1303-1315 (December, 2019). DOI: 10.11650/tjm/190105


The circular descent of a permutation $\sigma$ is a set $\{ \sigma(i) \mid \sigma(i) \gt \sigma(i+1) \}$. In this paper, we focus on the enumerations of permutations by the circular descent set. Let $\operatorname{cdes}_n(S)$ be the number of permutations of length $n$ which have the circular descent set $S$. We derive the explicit formula for $\operatorname{cdes}_n(S)$. We describe a class of generating binary trees $T_k$ with weights. We find that the number of permutations in the set $\operatorname{CDES}_n(S)$ corresponds to the weights of $T_k$. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.


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Hungyung Chang. Jun Ma. Jean Yeh. "Enumerations of Permutations by Circular Descent Sets." Taiwanese J. Math. 23 (6) 1303 - 1315, December, 2019.


Received: 19 November 2018; Revised: 10 January 2019; Accepted: 15 January 2019; Published: December, 2019
First available in Project Euclid: 28 January 2019

zbMATH: 07142974
MathSciNet: MR4033546
Digital Object Identifier: 10.11650/tjm/190105

Primary: 05A15

Keywords: circular descent , generating tree , permutation , permutation tableaux

Rights: Copyright © 2019 The Mathematical Society of the Republic of China


Vol.23 • No. 6 • December, 2019
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